# Explain this passage on the proof that the curvature of the vector function $r(t)$ is $\frac{|r'(t) \times r''(t)|}{ |r'(t)|³}$?

Explain this step on the proof of the curvature of the vector $r(t)$?

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Which part you don't understand? –  Jack Mar 22 '12 at 6:21
Differentiate $T\times T=0$ and you are done –  Blah Mar 22 '12 at 6:47
I don't get it... –  Dokkat Mar 24 '12 at 7:26

Observe that $r$ and $T$ are functions of the variable $t$, so $r'$ is. But $r'$ is a product of functions, therefore, using Leibniz' rule, you can derive it, obtaining $r''$. Now $T \times T$ is $0$ because, clearly, these vectors aren't linearly indipendent. Finally, calculate the wedge product in $\mathbb{R}^3$ of the vectors $r'$ and $r''$. By distributive law, using the fact that $T \times T=0$, you obtain the result stated.